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Lie Groups, Lie Algebras, and Representations: An Elementary Understanding

Brian C. Hall

Chapter 5

The Representations of $\mathrm{SU}(3)$ - all with Video Answers

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Chapter Questions

01:04

Problem 1

Show that the roots listed in (5.3) are the only roots.

Carson Merrill
Carson Merrill
Numerade Educator

Problem 2

Let $\pi$ be an irreducible finite-dimensional representation of $\mathrm{sl}(3 ; \mathbb{C})$ acting on a space $V$ and let $\pi^*$ be the dual representation to $\pi$, acting $\mathrm{cu} V^*$, as defined in Section 4.7. Show that the weights of $\pi^*$ are the negatives of the weights of $\pi$.
Hint: Choose a basis for $V$ in which both $\pi\left(H_1\right)$ and $\pi\left(H_2\right)$ are diagonal.

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Problem 3

As in Exercise 2, let $\pi$ be an irreducible representation of $\mathrm{sl}(3 ; \mathrm{C})$ and let $\pi^*$ be the dual representation to $\pi$. Show that if $\pi$ has highest weight ( $m_1, m_2$ ), $\pi^*$ has highest weight ( $m_2, m_1$ ).
Hint: Establish this first in the cases $\left(m_1, m_2\right)=(1,0)$ and $\left(m_1, m_2\right)=$ $(0,1)$.

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Problem 4

Consider the adjoint representation of $s l(3 ; C)$ as a representation of $s \mid(2 ; \mathrm{C})$ by restricting the adjoint representation to the subalgebra spanned by $X_1, Y_1$, and $H_1$. Decompose this representation as a direct sum of irreducible representations of $\mathrm{sl}(2 ; \mathrm{C})$. Which representations occur and with what multiplicity?

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Problem 5

Following the method of Section 5.5 , work out the representation of $\mathrm{sI}(3 ; \mathbb{C})$ with highest weight $(2,0)$, acting on a subspace of $\mathrm{C}^3 \otimes \mathbb{C}^3$. Determine all the weights of this representation and their multiplicity (i.e., the dimension of the corresponding weight space). Verify that the dimension formula (Theorem 5-10) holds in this case.

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Problem 6

Consider the nine-dimensional representation of $\mathbf{s l}(3 ; \mathbb{C})$ considered in Section 5.5 , namely the tensor product of the representations with highest weights $(1,0)$ and $(0,1)$. Decompose this representation as a direct sum of irreducibles. Do the same for the tensor product of two copies of the irreducible representation with highest weight ( 1,0 ). (Compare Exercise 5.)

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Problem 7

Let $V_m$ denote the space of homogeneous polynomials on $\mathbb{C}^3$ of degree $m$. By imitating Section 4.3, construct a representation of SU(3) acting on $V_{\mathrm{m}}$. Find the weights for the associated action of $\mathrm{sl}(3 ; \mathbb{C})$ on $V_1$ and $V_2$. Show that $V_1$ and $V_2$ are irreducible representations (of $\mathrm{SU}(3)$ or $\mathrm{sl}(3 ; \mathbb{C})$ ). What are the highest weights of these representations?

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Problem 8

Show that $Z$ and $N$ (defined in Definition 5.20) are subgroups of $\mathrm{SU}(3)$. Show that $Z$ is a normal subgroup of $N$.

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Problem 9

For each permutation $\sigma$ of $\{1,2,3\}$, let $A_\sigma$ be the matrix such that $A e_k=$ $\operatorname{sgn}(\sigma) e_{\sigma(k)}$, where $\operatorname{sgn}(\sigma)$ is the sign of the permutation $\sigma$, equal to 1 for even permutations and equal to -1 for odd permutations. Show that the matrices $A_\sigma$ form a subgroup of $N$ that is isomorphic to $W$.

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Problem 10

Consider the matrix $A$ in $\mathrm{SU}(3)$ given by

$$
A=\left(\begin{array}{lll}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array}\right)
$$

which maps $e_1$ to $e_2, e_2$ to $e_3$, and $e_3$ to $e_1$. Let $H$ be an arbitrary element of $\mathfrak{b}$ and let $\lambda_2, \lambda_2$, and $\lambda_3$ be the diagonal entries of $H$ (which must sum to zero). Compute by hand $A H A^{-1}$ and verify that this is related to $H$ as described in Section 5.6, namely that $\lambda_1$ gets shifted into the second spot, $\lambda_2$ gets shifted into the third spot, and $\lambda_3$ gets shifted into the first spot.

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Problem 11

Show that under the identification of $b^*$ with $b$ described in Section 5.6, the action of $W$ on $\mathfrak{h}^*$ (described in (5.16)) coincides with the adjoint action of $W$ on $\mathbf{b}$.

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Problem 12

Regard the Weyl group as a group of linear transformations of $\mathfrak{b}$. Show that $-I$ is not an element of the Weyl group. Which representations of $\mathrm{sl}(3 ; \mathbb{C})$ have the property that their weights are invariant under $-I$ ?

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Problem 13

Using the proof of Proposition 5.13, show that every weight $\mu$ of an irreducible representation with highest weight $\mu_0$ must satisfy Condition 2 of Theorem 5.26.

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Problem 14

This exercise asks one to "prove" geometrically the following result. Let $\mu_0$ be a dominant integral element and $\mu$ any integral element. If $w-\mu$ is lower than $\mu_0$ for all $w \in W$, then $\mu$ is contained in the convex hull of the $W$-orbit of $\mu_0$.
To see why this result is true, make a picture of a typical dominant integral element $\mu_0$ and its $W$-orbit. Now, take a typical point $\mu$ that is not in the convex hull of the orbit of $\mu_0$ and draw its $W$-orbit. Show that the $W$-orbit of $\mu$ contains at least one point that is not lower than $\mu_0$. This result (along with the invariance of the weights under the action of the Weyl group) shows that Condition 1 of Theorem 5.26 is a necessary condition for $\mu$ to be a weight of the representation with highest weight $\mu_0$.

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